Definition:Generating Function

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Definition

Let $A = \left \langle {a_n}\right \rangle$ be a sequence in $\R$.

Then $\displaystyle G_A \left({z}\right) = \sum_{n \ge 0} a_n z^n$ is called the generating function for the sequence $A$.

The mapping $G_A \left({z}\right)$ is defined for all $z$ for which the power series $\displaystyle \sum_{n \ge 0} a_n z^n$ is convergent.

The definition can be modified so that the lower limit of the summation is $b$ where $b > 0$ by assigning $a_k = 0$ where $0 \le k < b$.

When the sequence is understood, $G \left({z}\right)$ can be used.

Different authors use different symbols. $\zeta \left({z}\right)$ is sometimes seen but can be confused with the Riemann zeta function.

The variable is a dummy -- $x$ is often used instead.

A generating function is a clothesline on which we hang up a sequence of numbers for display.

Herbert Wilf, generatingfunctionology (1994)

Everybody else quotes it (it's the first line of the above book), so I don't see why this site should be any different.